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How did Bernhard Riemann generalize geometry to curved spaces of higher dimensions?

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differential geometry An annular strip (the region between two concentric circles) can be cut and bent into a helical strake that follows approximately the contour of a cylinder. Techniques of differential geometry are employed to find the dimensions of the annular strip that will best match the required curvature of the strake.

Bernhard Riemann revolutionized geometry by shifting the focus from the rigid constraints of Euclidean space to a more flexible, conceptual framework. In his seminal 1854 lecture, he proposed that the fundamental components of geometry are a space of points, now known as a manifold, and a method for measuring distances along curves within that space.1

Riemann argued that geometry need not be restricted to three dimensions or to the properties of ordinary Euclidean space; instead, he allowed for spaces of any dimension. By defining a way to measure distance—which later evolved into the use of the metrical tensor—he provided the tools to describe the intrinsic curvature of these higher-dimensional spaces.12

This approach moved geometry away from "constructive proofs" toward a more conceptual understanding, enabling the study of spaces that do not require an external embedding in a higher-dimensional environment. His work laid the essential mathematical foundation for Albert Einstein’s general theory of relativity, where gravity is understood as a manifestation of the curvature of four-dimensional space-time.12